{"status": "success", "data": {"description_md": "A unit cube has vertices $P_1,P_2,P_3,P_4,P_1',P_2',P_3',$ and $P_4'$. Vertices $P_2$, $P_3$, and $P_4$ are adjacent to $P_1$, and for $1\\le i\\le 4,$ vertices $P_i$ and $P_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $P_1P_2$, $P_1P_3$, $P_1P_4$, $P_1'P_2'$, $P_1'P_3'$, and $P_1'P_4'$. What is the octahedron's side length?<br><center><img class=\"problem-image\" alt=\"[asy] import three;  size(7.5cm); triple eye = (-4, -8, 3); currentprojection = perspective(eye);  triple[] P = {(1, -1, -1), (-1, -1, -1), (-1, 1, -1), (-1, -1, 1), (1, -1, -1)}; // P[0] = P[4] for convenience triple[] Pp = {-P[0], -P[1], -P[2], -P[3], -P[4]};  // draw octahedron triple pt(int k){ return (3*P[k] + P[1])/4; } triple ptp(int k){ return (3*Pp[k] + Pp[1])/4; } draw(pt(2)--pt(3)--pt(4)--cycle, gray(0.6)); draw(ptp(2)--pt(3)--ptp(4)--cycle, gray(0.6)); draw(ptp(2)--pt(4), gray(0.6)); draw(pt(2)--ptp(4), gray(0.6)); draw(pt(4)--ptp(3)--pt(2), gray(0.6) + linetype(&quot;4 4&quot;)); draw(ptp(4)--ptp(3)--ptp(2), gray(0.6) + linetype(&quot;4 4&quot;));  // draw cube for(int i = 0; i &lt; 4; ++i){ \tdraw(P[1]--P[i]); draw(Pp[1]--Pp[i]); \tfor(int j = 0; j &lt; 4; ++j){ \t\tif(i == 1 || j == 1 || i == j) continue; \t\tdraw(P[i]--Pp[j]); draw(Pp[i]--P[j]); \t} \tdot(P[i]); dot(Pp[i]); \tdot(pt(i)); dot(ptp(i)); }  label(&quot;$P_1$&quot;, P[1], dir(P[1])); label(&quot;$P_2$&quot;, P[2], dir(P[2])); label(&quot;$P_3$&quot;, P[3], dir(-45)); label(&quot;$P_4$&quot;, P[4], dir(P[4])); label(&quot;$P'_1$&quot;, Pp[1], dir(Pp[1])); label(&quot;$P'_2$&quot;, Pp[2], dir(Pp[2])); label(&quot;$P'_3$&quot;, Pp[3], dir(-100)); label(&quot;$P'_4$&quot;, Pp[4], dir(Pp[4])); [/asy]\" class=\"latexcenter\" height=\"342\" src=\"https://latex.artofproblemsolving.com/5/9/d/59d2570727a6deab28e0d8d2be5154efc56b50ca.png\" width=\"342\"/></center>\n\n$\\textbf{(A)}\\ \\frac{3\\sqrt{2}}{4}\\qquad\\textbf{(B)}\\ \\frac{7\\sqrt{6}}{16}\\qquad\\textbf{(C)}\\ \\frac{\\sqrt{5}}{2}\\qquad\\textbf{(D)}\\ \\frac{2\\sqrt{3}}{3}\\qquad\\textbf{(E)}\\ \\frac{\\sqrt{6}}{2}$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>A unit cube has vertices  <span class=\"katex--inline\">P_1,P_2,P_3,P_4,P_1',P_2',P_3',</span>  and  <span class=\"katex--inline\">P_4'</span> . Vertices  <span class=\"katex--inline\">P_2</span> ,  <span class=\"katex--inline\">P_3</span> , and  <span class=\"katex--inline\">P_4</span>  are adjacent to  <span class=\"katex--inline\">P_1</span> , and for  <span class=\"katex--inline\">1\\le i\\le 4,</span>  vertices  <span class=\"katex--inline\">P_i</span>  and  <span class=\"katex--inline\">P_i'</span>  are opposite to each other. A regular octahedron has one vertex in each of the segments  <span class=\"katex--inline\">P_1P_2</span> ,  <span class=\"katex--inline\">P_1P_3</span> ,  <span class=\"katex--inline\">P_1P_4</span> ,  <span class=\"katex--inline\">P_1'P_2'</span> ,  <span class=\"katex--inline\">P_1'P_3'</span> , and  <span class=\"katex--inline\">P_1'P_4'</span> . What is the octahedron&#8217;s side length?<br/><center><img class=\"latexcenter\" alt=\"[asy] import three;  size(7.5cm); triple eye = (-4, -8, 3); currentprojection = perspective(eye);  triple[] P = {(1, -1, -1), (-1, -1, -1), (-1, 1, -1), (-1, -1, 1), (1, -1, -1)}; // P[0] = P[4] for convenience triple[] Pp = {-P[0], -P[1], -P[2], -P[3], -P[4]};  // draw octahedron triple pt(int k){ return (3*P[k] + P[1])/4; } triple ptp(int k){ return (3*Pp[k] + Pp[1])/4; } draw(pt(2)--pt(3)--pt(4)--cycle, gray(0.6)); draw(ptp(2)--pt(3)--ptp(4)--cycle, gray(0.6)); draw(ptp(2)--pt(4), gray(0.6)); draw(pt(2)--ptp(4), gray(0.6)); draw(pt(4)--ptp(3)--pt(2), gray(0.6) + linetype(&#34;4 4&#34;)); draw(ptp(4)--ptp(3)--ptp(2), gray(0.6) + linetype(&#34;4 4&#34;));  // draw cube for(int i = 0; i &lt; 4; ++i){ &#9;draw(P[1]--P[i]); draw(Pp[1]--Pp[i]); &#9;for(int j = 0; j &lt; 4; ++j){ &#9;&#9;if(i == 1 || j == 1 || i == j) continue; &#9;&#9;draw(P[i]--Pp[j]); draw(Pp[i]--P[j]); &#9;} &#9;dot(P[i]); dot(Pp[i]); &#9;dot(pt(i)); dot(ptp(i)); }  label(&#34;$P_1$&#34;, P[1], dir(P[1])); label(&#34;$P_2$&#34;, P[2], dir(P[2])); label(&#34;$P_3$&#34;, P[3], dir(-45)); label(&#34;$P_4$&#34;, P[4], dir(P[4])); label(&#34;$P'_1$&#34;, Pp[1], dir(Pp[1])); label(&#34;$P'_2$&#34;, Pp[2], dir(Pp[2])); label(&#34;$P'_3$&#34;, Pp[3], dir(-100)); label(&#34;$P'_4$&#34;, Pp[4], dir(Pp[4])); [/asy]\" height=\"342\" src=\"https://latex.artofproblemsolving.com/5/9/d/59d2570727a6deab28e0d8d2be5154efc56b50ca.png\" width=\"342\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ \\frac{3\\sqrt{2}}{4}\\qquad\\textbf{(B)}\\ \\frac{7\\sqrt{6}}{16}\\qquad\\textbf{(C)}\\ \\frac{\\sqrt{5}}{2}\\qquad\\textbf{(D)}\\ \\frac{2\\sqrt{3}}{3}\\qquad\\textbf{(E)}\\ \\frac{\\sqrt{6}}{2}</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2012 AMC 12B Problem 19", "can_next": true, "can_prev": true, "nxt": "/problem/12_amc12B_p20", "prev": "/problem/12_amc12B_p18"}}